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A243662 Triangle read by rows: the reversed x = 1+q Narayana triangle at m=2. 3
1, 3, 1, 12, 8, 1, 55, 55, 15, 1, 273, 364, 156, 24, 1, 1428, 2380, 1400, 350, 35, 1, 7752, 15504, 11628, 4080, 680, 48, 1, 43263, 100947, 92169, 41895, 9975, 1197, 63, 1, 246675, 657800, 708400, 396704, 123970, 21560, 1960, 80, 1, 1430715, 4292145, 5328180, 3552120, 1381380, 318780, 42504, 3036, 99, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

See Novelli-Thibon (2014) for precise definition.

LINKS

Table of n, a(n) for n=1..55.

J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014. See Fig. 10.

FORMULA

T(n,k) = (binomial(3*n+1,n) * binomial(n,k-1) * binomial(n-1,k-1)) / (binomial(3*n,k-1) * (3*n+1)) = (A001764(n) * A001263(n,k) * k) / binomial(3*n,k-1) for 1 <= k <= n (conjectured). - Werner Schulte, Nov 22 2018

T(n,k) = binomial(3*n+1-k,n-k) * binomial(n,k-1) / n for 1 <= k <= n, more generally: T_m(n,k) = binomial((m+1)*n+1-k,n-k) * binomial(n,k-1) / n for 1 <= k <= n and some fixed integer m>1. - Werner Schulte, Nov 22 2018

EXAMPLE

Triangle begins:

     1;

     3,    1;

    12,    8,    1;

    55,   55,   15,   1;

   273,  364,  156,  24,  1;

  1428, 2380, 1400, 350, 35, 1;

  ...

MATHEMATICA

T[m_][n_, k_] := Binomial[(m + 1) n + 1 - k, n - k] Binomial[n, k - 1]/n;

Table[T[2][n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-Fran├žois Alcover, Feb 12 2019 *)

CROSSREFS

Cf. A001764, A001263, A243663 (m=3).

Row sums give A003168.

Row reversed triangle is A102537.

Sequence in context: A133366 A049458 A143492 * A062139 A156366 A144353

Adjacent sequences:  A243659 A243660 A243661 * A243663 A243664 A243665

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Jun 13 2014

EXTENSIONS

Data and Example (T(2,2) and T(5,3)) corrected and more terms added by Werner Schulte, Nov 22 2018

STATUS

approved

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Last modified October 29 19:06 EDT 2020. Contains 338067 sequences. (Running on oeis4.)